Question

Find dy/dx by implicit differentiation

cos(xy²)= y

Answer 1

Given

`\cos(xy^2) = y`

Chain rule:

`(d\cos(xy^2))/(dx) = (dy)/(dx)`

`-\sin(xy^2) (d(xy^2))/(dx) = (dy)/(dx)`

Product rule:

`-\sin(xy^2) \left(x(dy^2)/(dx) + y^2 (dx)/(dx)\right) = (dy)/(dx)`

`-\sin(xy^2) \left(2xy(dy)/(dx) + y^2\right) = (dy)/(dx)`

Solve for
`(dy)/(dx)`.

`-2xy\sin(xy^2) (dy)/(dx) - y^2 \sin(xy^2) = (dy)/(dx)`

`\left(1+2xy\sin(xy^2)\right) (dy)/(dx) = -y^2 \sin(xy^2)`

`(dy)/(dx) = \boxed{-(y^2 \sin(xy^2))/(1 + 2xy \sin(xy^2))}`